Question

1) What is the derivative of e^{​​​​​​y} with respect to x?

2) if cos y =xcos(a+y), with cos a not equal to +_1. Prove that dy/dx = cos^{​​​​2}(a+y)/sin a.

Answer 1

[1]
First take the derivative like you normally would:

e^y

Then take the derivative of the stuff substituted inside, the stuff where an x would usually be:

(d/dx)y= dy/dx

Multiply them together.

e^y * dy/dx

Summarized mathemetically,

du/dx = du/dy * dy/dx

where here u=e^y



[2]

Given : cos y = x · cos (y+a) ......... (1)

∴ x = cos y / cos (y+a)

∴ dx/dy

= [ cos (y+a). ( - sin y ) - ( cos y ). ( - sin (y+a)) ] / cos² (y+a)

= [ sin (y+a). cos y - cos (y+a). sin y ] / cos² (y+a)

= { sin [ (y+a) - y ] } / cos² (y+a)

= sin a / cos² (y+a).

∴ dy/dx = [ 1 / ( dx/dy ) ] = cos² (y+a) / sin a